Driving Euler-Bernoulli equation for a compressible viscous flow from continuum conservation equations of an steady-state flow

My understanding of Euler-Bernoulli (EB) equation is that it is the energy balance equation in Lagrangian form for an inviscid, incompressible and steady-state flow . From different sources I see different equations attempting towards extending the equation for compressible or viscous flows. For example the wikipedia entry for the matter offers:

$$v dv+frac{dP}{rho}=0 tag{1}$$

along the stream for compressible flow (assuming there is no body force i.e. gravity) and I have also seen:

$$v dv+ dh=0 tag{2}$$

also along the stream for viscous flows (which paradoxically does not include viscosity).

If my understanding of the EB equation is correct then we must be able to drive them from Eulerian form of continuum equation of energy balance neglecting conduction and radiation:

$$
left{
begin{matrix}
rho left( check{v} check{nabla}^T right) e=check{sigma} : check{nabla}^T check{v}
left( check{v} check{nabla}^T right) left( h+frac{v^2}{2} right)=0
end{matrix}right. tag{3}$$

Where

$$ check{sigma}= check{tau} -Pcheck{I} tag{4}$$

and

$$check{tau}=etaleft(
check{nabla}^T check{v}+
left(
check{nabla}^T check{v}
right)^T
right)
+lambda
left(check{nabla}check{v}^Tright)
check{I} tag{5}$$

For Newtonian fluids.

So my questions are:

1. Am I right about extended EB equations being Lagrangian form of the energy balance equation for steady state flow?
2. If no then how these two distinct set of equations are related? the EB equation and continuum equations for conservation of mass (i.e. continuity), momentum (i.e. Navier-Stokes) and energy.
3. If yes how we can drive the extended EB equation for compressible viscous flow from eq 3 and what is the correct form?